I need to either solve the following Diophantine equation with three unknowns:
$n_{1} (n_{1} +1) \pm n_{2} (n_{2} +1) \pm n_{3} (n_{3} +1) = 0$,
where $n_{1,2,3}$ can be positive or negative -- or perhaps prove that this equation doesn't have a solution other than the trivial one
$n_{1} = 0,\hspace{5mm} n_{2} = n_{3}$
plus the obvious permutations.
Given that this equation looks pretty much like the one in the Fermat Theorem, I personally think that the latter is more plausible.
On
Ummm. Since $n(n+1) \geq 0,$ it is necessary to have final form $$ n_1(n_1 + 1) = n_2(n_2 + 1) + n_3(n_3 + 1) $$ to get anything worthwhile. Multiply through by $4,$ add $2$ to both sides. The result is $$ x^2 + 1 = z^2 + w^2, $$ with $x,z,w$ odd. The solutions to this, all of them, are parametrized by the modular group. take integers $a,b,c,d$ with $$ ad - bc = 1. $$ Then take $$ x = ab + cd; \; \; z = ab - cd; \; \; w = ad + bc $$ with the restriction the $x,z,w$ must be odd. Then let $$ n_1 = \frac{x-1}{2}, \; \; n_2 = \frac{z-1}{2}, \; \; n_3 = \frac{w-1}{2}. $$ This means, with $ad-bc = 1,$ $$ n_1 = \frac{ab + cd-1}{2}, \; \; n_2 = \frac{ab - cd-1}{2}, \; \; n_3 = \frac{ad + bc-1}{2}. $$
If we do not restrict $y,$ just say $x^2 + y^2 = z^2 + w^2,$ with $x$ largest and $x \equiv z \pmod 2,$ we can parametrize all solutions in the same manner, just add $y = ad - bc.$ To get all entries positive we can even take $a > c > 0,$ $b > d > 0,$ and $ad \geq bc.$ I had a certain amount of repetition removed from this printout. There are some repeats left.
jagy@phobeusjunior:~$ ./two_pair | sort -n
x^2 + y^2 = z^2 + w^2
x y z w a b c d
5 0 3 4 2 2 1 1
7 1 5 5 3 2 1 1
8 1 4 7 2 3 1 2 // a > c ; b > d ;
9 2 7 6 4 2 1 1 // a d >= b c
11 2 5 10 2 4 1 3 // x = a * b + c * d
11 3 9 7 5 2 1 1 // y = a * d - b * c
12 1 8 9 5 2 2 1 // z = a * b - c * d
13 0 5 12 3 3 2 2 // w = a * d + b * c
13 1 11 7 4 3 1 1
13 4 11 8 6 2 1 1
14 3 6 13 2 5 1 4
14 5 10 11 4 3 1 2
15 5 13 9 7 2 1 1
16 3 12 11 7 2 2 1
17 0 15 8 4 4 1 1
17 1 13 11 3 5 1 2
17 4 7 16 2 6 1 5
17 6 15 10 8 2 1 1
17 7 13 13 5 3 1 2
18 1 10 15 2 7 1 4
18 1 6 17 3 4 2 3
19 2 13 14 8 2 3 1
19 3 17 9 6 3 1 1
19 4 11 16 5 3 2 2
19 7 17 11 9 2 1 1
19 8 13 16 4 4 1 3
20 5 16 13 9 2 2 1
20 5 8 19 2 7 1 6
20 9 16 15 6 3 1 2
21 1 19 9 5 4 1 1
21 2 11 18 2 8 1 5
21 8 19 12 10 2 1 1
x y z w a b c d
x^2 + y^2 = z^2 + w^2
If $n$ is an integer, no matter what its sign is, then $n(n-1)$ will be either zero or positive, with zero exactly if $n=0$ or $n=1$.
Since neither of the terms in your equation can be negative, the only way for the entire thing to be zero is if each term is zero, so you have exactly $8$ solutions, corresponding to each combination of $n_1,n_2,n_3\in\{0,1\}$.
The numbers of the form $n(n+1)$ are $$ 0, 2, 6, 12, 20, 30, \ldots $$ Their differences are the successive even numbers, and since all the numbers are themselves even, we can just pick one of them and express it as the difference of two others, for example $$ 3(3+1) + 5(5+1) - 6(6+1) = 0 $$ and in general we have solutions of the form $$ n_1 \in \mathbb Z, \qquad n_3 = \frac{n_1(n_1+1)}2, \qquad n_2 = n_3-1 $$ This doesn't exhaust the solutions; we also have, for example $$ 5(5+1) + 6(6+1) - 8(8+1) = 0 $$ $$ 9(9+1) + 13(13+1) - 16(16+1) = 0 $$ $$ 11(11+1) + 14(14+1) - 18(18+1) = 0 $$